About Robustness of Fuzzy Logic PD and PID Controller under Changes of Reasoning Methods

In the paper fuzzy controllers with if ... then type of rules are considered. Each rule can contain linguistic values joint by sentence conjunctions “and”, “or”. Rules are jointed by word “also”. Different mathematical interpretations are used for “and”, “or”, “also” during approximate reasoning. Rules with same consequences can be aggregated in one rule or not aggregated. Moreover, different defuzzification methods are applied to obtain crisp output value of the controllers. Dynamic of the system with linear or non-linear plants was investigated. Very interesting fact was observed. The system dynamic is practically unchanged if triangular norms or B-operations (definition is in the paper) are used for interpretation of “and”, “or”, “also”, and typical defuzzification methods as height, areas, gravity, center of area, center of gravity, Tsukamoto are applied. “Fuzzy theorem” concerning dynamic of the system is presented.